Multiple regression principles

Notes and in-class exercises

Notes

Learning goals

Working with multiple predictors in our plots and models can get complicated! There are no recipes for this process. BUT there are some guiding principles that assist in long-term retention, deeper understanding, and the ability to generalize our tools in new settings. By the end of this lesson, you should be familiar with some general principles for…

  • incorporating additional quantitative or categorical predictors in a visualization
  • how additional quantitative or categorical predictors impact the physical representation of a model
  • interpreting quantitative or categorical coefficients in a multiple regression model

Readings and videos

Please watch the following video before class.

Exercises

Let’s revisit the bikeshare data:

# Load packages & import data
library(readr)
library(ggplot2)
library(dplyr)

bikes <- read_csv("https://mac-stat.github.io/data/bikeshare.csv") %>% 
  rename(rides = riders_registered)

Our goal is to understand how / why registered ridership from day to day. To this end, we’ll build various multiple linear regression models of rides by different combinations of the possible predictors:

# Check out the data
head(bikes)
# A tibble: 6 × 15
  date       season  year month day_of_week weekend holiday temp_actual
  <date>     <chr>  <dbl> <chr> <chr>       <lgl>   <chr>         <dbl>
1 2011-01-01 winter  2011 Jan   Sat         TRUE    no             57.4
2 2011-01-02 winter  2011 Jan   Sun         TRUE    no             58.8
3 2011-01-03 winter  2011 Jan   Mon         FALSE   no             46.5
4 2011-01-04 winter  2011 Jan   Tue         FALSE   no             46.8
5 2011-01-05 winter  2011 Jan   Wed         FALSE   no             48.7
6 2011-01-06 winter  2011 Jan   Thu         FALSE   no             47.1
# ℹ 7 more variables: temp_feel <dbl>, humidity <dbl>, windspeed <dbl>,
#   weather_cat <chr>, riders_casual <dbl>, rides <dbl>, riders_total <dbl>

Exercise 1: Review visualization

Let’s build a model of rides by windspeed (quantitative) and weekend status (categorical). Plot & describe, in words, the relationship between these 3 variables. Include a representation of the regression model.

# Plot of rides vs windspeed & weekend
# HINT: Start with a plot of rides vs windspeed

Exercise 2: Review model

Let’s build the model:

bike_model_1 <- lm(rides ~ windspeed + weekend, data = bikes)
coef(summary(bike_model_1))
              Estimate Std. Error   t value      Pr(>|t|)
(Intercept) 4738.38053  147.53653 32.116659 1.208405e-141
windspeed    -63.97072   10.45274 -6.119997  1.528443e-09
weekendTRUE -925.15701  119.86330 -7.718434  3.891082e-14

The model formula thus is:

E[rides | windspeed, weekendTRUE] = 4738.38 - 63.97 * windspeed - 925.16 * weekendTRUE

This model formula is represented by 2 lines, one corresponding to weekends and the other to weekdays. Specify the corresponding sub-model formulas:

weekdays: rides = ___ - ___ windspeed

weekends: rides = ___ - ___ windspeed

Exercise 3: Review coefficient interpretation

  1. The intercept coefficient, 4738.38, represents the intercept of the sub-model for weekdays, the reference category. What’s its contextual interpretation?

  2. The windspeed coefficient, -63.97, represents the shared slope of the weekend and weekday sub-models. What’s its contextual interpretation?

  3. The weekendTRUE coefficient, -925.16, represents the change in intercept for the weekend vs weekday sub-model. What’s its contextual interpretation?

Exercise 4: 2 categorical predictors – visualization

Thus far, we’ve explored a couple examples of multiple regression models that have 2 predictors, 1 quantitative and 1 categorical. So what happens when both predictors are categorical?! To this end, let’s model rides by weekend status and season.

The below code plots rides vs season. Modify this code to also include information about weekend. HINT: Remember the visualization principle that additional categorical predictors require some sort of grouping mechanism / mechanism that distinguishes between the 2 groups.

# rides vs season
bikes %>% 
  ggplot(aes(y = rides, x = season)) + 
  geom_boxplot()

# rides vs season AND weekend
# bikes %>%
#   ggplot(aes(y = rides, x = season, ___ = ___)) + 
#   geom_boxplot()

Exercise 5: follow-up

  1. Describe (in words) the relationship of ridership with season & weekend status.

  2. A model of rides by season alone would be represented by only 4 expected outcomes, 1 for each season. Considering this and the plot above, how do you anticipate a model of rides by season and weekend status will be represented?

    • 2 lines, 1 for each weekend status
    • 8 lines, 1 for each possible combination of season & weekend
    • 2 expected outcomes, 1 for each weekend status
    • 8 expected outcomes, 1 for each possible combination of season & weekend

Exercise 6: 2 categorical predictors – build the model

Let’s build the multiple regression model of rides vs season and weekend:

bike_model_2 <- lm(rides ~ weekend + season, bikes)
coef(summary(bike_model_2))
               Estimate Std. Error     t value      Pr(>|t|)
(Intercept)   4260.4492   99.16363  42.9638294 1.384994e-201
weekendTRUE   -912.3324  103.23016  -8.8378473  7.298199e-18
seasonspring  -116.3824  132.76018  -0.8766364  3.809741e-01
seasonsummer   438.4424  132.06413   3.3199205  9.454177e-04
seasonwinter -1719.0572  133.30505 -12.8956646  2.081758e-34

Thus the model formula is

E[rides | weekend, season] = 4260.45 - 912.33 weekendTRUE - 116.38 seasonspring + 438.44 seasonsummer - 1719.06 seasonwinter

  1. Use this model to predict the ridership on the following days:
# a fall weekday
# 4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___

# a winter weekday    
# 4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___

# a fall weekend day        
# 4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___

# a winter weekend day
# 4260.45 - 912.33*___ - 116.38*___  + 438.44*___ - 1719.06*___
  1. We only made 4 predictions here. How many possible predictions does this model produce? Is this consistent with your intuition in the previous exercise?

Exercise 7: 2 categorical predictors – interpret the model

Use your above predictions and visualization to fill in the below interpretations of the model coefficients. Hint: What is the consequence of plugging in 0 or 1 for the different weekend and season categories?

  1. Interpreting 4260: We expect there to be 4260 riders on (weekdays/weekends) during the (fall/spring/summer/winter).

  2. Interpreting -912: In any season, we expect there to be 912 (more/fewer) riders on weekends than on ___.

  3. Interpreting -1719: On both weekdays and weekends, we expect there to be 1719 (more/fewer) riders in winter than in ___.

Exercise 8: 2 quantitative predictors – visualization

Next, consider the relationship between rides and 2 quantitative predictors: windspeed and temp_feel. Check out 2 plots of this relationship below. These reflect the visualization principle that quantitative variables require some sort of numerical scaling mechanism – rides and windspeed get numerical axes, and temp_feel gets size and color scales.

Modify the code below to recreate these 2 plots.

# Left plot
# bikes %>% 
#   ggplot(aes(y = rides, x = windspeed, ___ = ___)) + 
#   geom_point()

# Right plot
# bikes %>% 
#   ggplot(aes(y = rides, x = windspeed, ___ = ___)) + 
#   geom_point()

With some new code in the plotly (not ggplot2) package, we can also plot these 3 quantitative variables in 3D, each getting their own axis! First, enter install.packages("plotly") in the console (not Rmd). Then play around with this code:

library(plotly)

plot_ly(data = bikes,
        z = ~rides, y = ~windspeed, x = ~temp_feel, 
        type = "scatter3d", mode = "markers", size = 0.5)

Exercise 9: follow-up

  1. Describe (in words) the relationship of ridership with windspeed & temperature.

  2. A model of rides by windspeed alone would be represented by 1 line. How do you anticipate a model of rides by windspeed and temp_feel status will be represented? HINT: Think about the 3D scatterplot.

Exercise 10: 2 quantitative predictors – modeling

Let’s build the multiple regression model of rides vs windspeed and temp_feel:

bike_model_3 <- lm(rides ~ windspeed + temp_feel, bikes)
coef(summary(bike_model_3))
             Estimate Std. Error     t value     Pr(>|t|)
(Intercept) -24.06464 299.303032 -0.08040225 9.359394e-01
windspeed   -36.54372   9.408116 -3.88427585 1.119805e-04
temp_feel    55.51648   3.330739 16.66791759 4.436963e-53

Thus the model formula is

E[rides | windspeed, temp_feel] = -24.06 - 36.54 windspeed + 55.52 temp_feel

This is the formula of a plane, drawn here:

  1. Interpret the intercept coefficient, -24.06, in context.

  2. Interpret the windspeed coefficient, -36.54, in context. HINT: This is the slope of the red lines in the right plot, drawn at 3 arbitrary temperatures (40, 60, and 90 degrees).

  3. Interpret the temp_feel coefficient, 55.52, in context. HINT: This is the slope of the red lines in the left plot, drawn at 3 arbitrary wind speeds (5, 20, and 30 mph).



Exercise 11: Which is best?

We’ve now observed 3 different models of ridership, each having 2 predictors. The R-squared values of these models, along with those of the simple linear regression models with each predictor alone, are summarized below.

model predictors R-squared
bike_model_1 windspeed & weekend 0.119
bike_model_2 weekend & season 0.349
bike_model_3 windspeed & temp_feel 0.310
bike_model_4 windspeed 0.047
bike_model_5 temp_feel 0.296
bike_model_6 weekend 0.074
bike_model_7 season 0.279

  1. Which model does the best job of explaining the variability in ridership from day to day?

  2. If you could only pick one predictor, which would it be?

  3. What happens to R-squared when we add a second predictor to our model, and why does this make sense? For example, how does the R-squared for model 1 (with both windspeed and weekend) compare to those of model 4 (only windspeed) and model 6 (only weekend)?

  4. Are 2 predictors always better than 1? Provide evidence and explain why this makes sense.



Exercise 12: Principles of interpretation

These exercises have revealed some principles behind interpreting model coefficients, summarized below. Review and confirm that these make sense.

Principles of interpretation

Consider a multiple linear regression model:

\[E[y | x_1, x_2, ..., x_p] = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_p x_p\]

We can interpret the coefficients as follows:

  • \(\beta_0\) (“beta 0”) is the y-intercept. It describes the typical value of \(y\) when \(x_1, x_2,..., x_k\) are all 0, ie. when all quantitative predictors are set to 0 and the categorical predictors are set to their reference levels.

  • \(\beta_i\) (“beta i”) is the coefficient of \(x_i\).

    • If \(x_i\) is quantitative, \(\beta_i\) describes the typical change in \(y\) per 1-unit increase in \(x_i\) while at a fixed set of the other \(x\).
    • If \(x_i\) represents a category of a categorical variable, \(\beta_i\) describes the typical change in \(y\) when we move to this category from the reference category while at a fixed set of the other \(x\).





Extra practice

The following exercises provide extra practice. If you don’t get to these during class, you’re encouraged to try them outside class.

Exercise 13: Practice 1

Consider the relationship of rides vs weekend and weather_cat.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret the first 3 model coefficients.

Exercise 14: Practice 2

Consider the relationship of rides vs temp_feel and humidity.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret the first 3 model coefficients.

Exercise 15: Practice 3

Consider the relationship of rides vs temp_feel and weather_cat.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret the first 3 model coefficients.

Exercise 16: CHALLENGE

We’ve explored models with 2 predictors. What about 3 predictors?! Consider the relationship of rides vs temp_feel, humidity, AND weekend.

  1. Construct a visualization of this relationship.
  2. Construct a model of this relationship.
  3. Interpret each model coefficient.
  4. If you had to draw the shape of the model trend, what would it look like: a line, parallel lines, a plane, parallel planes?

Solutions

Exercise 1: Review visualization

bikes %>% 
  ggplot(aes(y = rides, x = windspeed, color = weekend)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE)
`geom_smooth()` using formula = 'y ~ x'

Exercise 2: Review model

weekdays: rides = 4738.38 - 63.97 windspeed

weekends: rides = 4738.38 - 63.97 windspeed - 925.16 = 3813.22 - 63.97 windspeed

Exercise 3: Review coefficient interpretation

  1. We expect roughly 4738 riders on weekdays with 0 windspeed.

  2. On both weekends and weekdays, we expect 64 fewer riders for every 1mph increase in windspeed.

  3. At any windspeed, we expect 925 fewer riders on weekend days than on weekdays.

Exercise 4: 2 categorical predictors – visualization

bikes %>% 
  ggplot(aes(y = rides, x = season, fill = weekend)) + 
  geom_boxplot()

Exercise 5: follow-up

  1. In every season, ridership tends to be lower on weekends. Across weekend status, ridership tends to be highest in summer and lowest in winter.

  2. 8 expected outcomes

Exercise 6: 2 categorical predictors – build the model

#fall weekday:    
4260.45 - 912.33*0 - 116.38*0 + 438.44*0 - 1719.06*0
[1] 4260.45
#winter weekday:
4260.45 - 912.33*0 - 116.38*0 + 438.44*0 - 1719.06*1
[1] 2541.39
#fall weekend:    
4260.45 - 912.33*1 - 116.38*0 + 438.44*0 - 1719.06*0
[1] 3348.12
#winter weekend:
4260.45 - 912.33*1 - 116.38*0 + 438.44*0 - 1719.06*1
[1] 1629.06
  1. 8: 2 weekend categories * 4 season categories

Exercise 7: 2 categorical predictors – interpret the model

  1. We expect there to be 4260 riders on weekdays during the fall.
  2. In any season, we expect there to be 912 fewer riders on weekends than on weekdays.
  3. On both weekdays and weekends, we expect there to be 1719 fewer riders in winter than in fall.

Exercise 8: 2 quantitative predictors – visualization

bikes %>% 
  ggplot(aes(y = rides, x = windspeed, size = temp_feel)) + 
  geom_point() 

bikes %>% 
  ggplot(aes(y = rides, x = windspeed, color = temp_feel)) + 
  geom_point()

Exercise 9: follow-up

Ridership tends to increase with temperature (no matter the windspeed) and decrease with windspeed (no matter the temperature).

Exercise 10: 2 quantitative predictors – modeling

  1. -24.06 = typical ridership on days with 0 windspeed and a 0 degree temperature. (This doesn’t make contextual sense, but indicates where the plane “lives” in space!)

  2. No matter the temperature, we expect 37 fewer riders for every 1mph increase in windspeed.

  3. No matter the windspeed, we expect 56 more riders for every 1 degree increase in temperature.

Exercise 11: Which is best?

  1. model 2
  2. temperature
  3. R-squared increases (our model is stronger when we include another predictor)
  4. nope. model 1 (with windspeed and weekend) has a lower R-squared than model 5 (with only temperature)



Exercise 13: Practice 1

bikes %>% 
  ggplot(aes(y = rides, x = weekend, fill = weather_cat)) + 
  geom_boxplot()

new_model_1 <- lm(rides ~ weekend + weather_cat, bikes)
coef(summary(new_model_1))
                    Estimate Std. Error   t value      Pr(>|t|)
(Intercept)        4211.8741   75.54724 55.751529 9.461947e-265
weekendTRUE        -982.2106  117.24719 -8.377264  2.786301e-16
weather_catcateg2  -608.8640  113.00211 -5.388077  9.628947e-08
weather_catcateg3 -2360.2049  319.71640 -7.382183  4.270163e-13

  • The typical ridership on a weekday with nice weather (categ1) is 4212 rides.

  • On days with similar weather, the typical ridership on a weekend is 982 rides less than on weekdays.

  • On similar days of the week, the typical ridership when the weather is “dreary” (categ2) is 609 rides less than when the weather is nice.



Exercise 14: Practice 2

bikes %>% 
  ggplot(aes(y = rides, x = temp_feel, color = humidity)) + 
    geom_point()

new_model_2 <- lm(rides ~ temp_feel + humidity, bikes)
coef(summary(new_model_2))
               Estimate Std. Error   t value     Pr(>|t|)
(Intercept)   315.83704 303.777334  1.039699 2.988249e-01
temp_feel      60.43316   3.272315 18.468015 9.451345e-63
humidity    -1868.99356 336.963661 -5.546573 4.078901e-08

  • It doesn’t really make sense to interpret the intercept – we didn’t see any days that were 0 degrees with 0 humidity.

  • On days with similar humidity, the typical ridership increases by 60 rides for each 1 degree increase in temperature.

  • On days with similar temperature, the typical ridership decreases by 1867 * 0.1 = 187 rides for each 0.1 increase in humidity levels.



Exercise 15: Practice 3

new_model_3 <- lm(rides ~ temp_feel + weather_cat, bikes)
coef(summary(new_model_3))
                     Estimate Std. Error   t value     Pr(>|t|)
(Intercept)        -288.68840 251.264383 -1.148943 2.509574e-01
temp_feel            55.30133   3.215495 17.198387 7.082670e-56
weather_catcateg2  -386.42241 100.187725 -3.856984 1.249775e-04
weather_catcateg3 -1919.01375 283.022420 -6.780430 2.481218e-11
bikes %>% 
  ggplot(aes(y = rides, x = temp_feel, color = weather_cat)) + 
  geom_point() + 
  geom_line(aes(y = new_model_3$fitted.values), size = 1.5)
Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

  • It doesn’t really make sense to interpret the intercept – we didn’t see any days that were 0 degrees.

  • On days with similar weather, the typical ridership increases by 55 rides for each 1 degree increase in temperature.

  • On days with similar temperature, the typical ridership is 386 rides lower on a dreary weather day (categ2) than a nice weather day (categ1).

Exercise 16: CHALLENGE

bikes %>% 
  ggplot(aes(y = rides, x = temp_feel, color = weekend, size = humidity)) + 
  geom_point()

new_model_4 <- lm(rides ~ temp_feel + humidity + weekend, bikes)
coef(summary(new_model_4))
               Estimate Std. Error   t value     Pr(>|t|)
(Intercept)   668.60236 292.181063  2.288315 2.240530e-02
temp_feel      59.36751   3.119256 19.032585 7.626695e-66
humidity    -1906.43437 320.982938 -5.939364 4.433789e-09
weekendTRUE  -869.05771 100.057822 -8.685555 2.471050e-17
  • It doesn’t really make sense to interpret the intercept – we didn’t see any days that were 0 degrees.
  • On days with similar humidity and time of week, the typical ridership increases by 59 rides for each 1 degree increase in temperature.
  • On days with similar temperature and time of week, the typical ridership decreases by 1906*0.1 = 190.6 rides for each 0.1 point increase in humidity levels.
  • On days with similar temperature and humidity, the typical ridership is 869 rides lower on weekends vs weekdays.

This model would look like 2 parallel planes, one for weekends and one for weekdays! Why? rides vs temp_feel (quant) would be a line. Adding in humidity (quant) would turn the model into a plane. Adding in weekend (cat) would split this one plane into two unique planes, one for each weekend category!